Microlocalization
of Topological
Boundary
Value
Morphism
and Regular-Specializable
Systems
Susumu
YAMAZAKI
(
山崎 晋
)*
Graduate School of Mathematical Sciences, the University ofTokyo,
8-1 Komaba3-chome, MegurO-ku, Tokyo 153-8914, Japan
Introduction
In microlocal analysis, it is
one
of the main subjects to give an appropriate formulationofthe boundary value problems for hyperfunction
or
microfunction solutions to asystemof linear partial differential equations with analytic coefficients (that is, acoherent (left)
$\mathcal{D}$-Module, here in this article,
we
shall write Module with acapital letter, instead ofsheaf of
modules). If the system is regular-specializable, the nearby-cycle of the systemcan
be defined in the theory of $\mathcal{D}$-Modules. After the results by Kashiwara and Os-hima [K-O], OsOs-hima [Os] and Schapira [Sc2], [Sc3], for any hyperfunction solutions toregular-specializable system Monteiro Fernandes [MF1] defined aboundary value
mor-phism which takes values in hyperfunction solutions to the nearby-cycle of the system
instead of the induced system. This morphism is injective (cf. [MF2]) and
ageneral-ization of the non-characteristic boundary value morphism (for the non-characteristic
case,
see
Komatsu and Kawai [KO-K], Schapira [Sc 1] and further Kataoka [Kat]$)$.More-over
recently Laurentand Monteiro Fernandes [L-MF2] reformulated this boundary value morphism and discussed the solvability underakind of hyperbolicitycondition (thenear-hyperbolicity). However, since this morphism is defined only for hyperfunction solutions,
amicrolocal boundary value problem is not considered. Therefore in this article, we
$5\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{l}$ state amicrolocalization oftheir result in the framework of Oaku [Oa2] and Oaku
Yamazaki [O-Y].
The details of this article will be given in
our
forthcoming paper [Y].ResearchFellow of The Japan Society for The Promotion ofScience
数理解析研究所講究録 1211 巻 2001 年 86-95
1Notation
We denote the set of integers, of real numbers and of complex numbers by $\mathbb{Z}$, $\mathbb{R}$ and $\mathbb{C}$
respectively as usual. Moreover we set $\mathrm{N}:=\{n\in \mathbb{Z};n\geq 1\}$ and $\mathrm{N}_{0}:=\mathrm{N}\mathrm{U}\{0\}$.
All the manifolds
are
assumed to be paracompact. Let $\tau:Earrow Z$ avector bundleover amanifold $Z$. Then, set $\dot{E}:=E\backslash Z$ and $\dot{\tau}$ the restriction of$\tau$ to $\dot{E}$
.
Let $M$ be
an
$(n+1)$-dimensional real analyticmanifold and $N$ aone-codimensional closed real analytic
submanifold of$M$. Let $X$ and $\mathrm{Y}$ be complexifications of$M$ and $N$ respectively such that $\mathrm{Y}$ is aclosed submanifold of$X$ and that $\mathrm{Y}\cap M=N$. Moreover, we
assume
the existenceofapartial complexification of $M$ in $X$;that is, there exists a $(2n+1)$-dimensional real
analytic submanifold $L$ of $X$ containing both $M$ and $\mathrm{Y}$ such that the triplet $(N, M, L)$
is locally isomorphic to $(\mathbb{R}^{n}\cross\{0\}, \mathbb{R}^{n+1}, \mathbb{C}^{n}\cross \mathbb{R})$ by alocal coordinate system $(z, \tau)=$
$(x+\sqrt{-1}y, t+\sqrt{-1}s)$ of $X$ around each point of $N$
.
We say such acoordinate systemadmissible. We shall mainly follow the notation in Kashiwara-Schapira [K-S]; we denote
the normal deformations of$N$ and $\mathrm{Y}$ in $M$ and $L$ by $\overline{M}_{N}$ and $\tilde{L}_{\mathrm{Y}}$ respectively and regard
$\overline{M}_{N}$ as aclosed submanifold of$\tilde{L}_{\mathrm{Y}}$ . We have the following commutative diagram:
and by admissible coordinates we have locally the following relation:
$N=\mathbb{R}_{x}^{n}\mathrm{J}|\cross\{0\}arrowarrow M=\mathbb{R}_{x}^{n}i\mathrm{J}|\cross$
$\mathrm{Y}=\mathbb{C}_{z}^{n}\cross\{0\}\frac{\tau i_{Y}}{\prime}L=\mathbb{C}_{z}^{n}\cross$ $=\mathbb{C}_{z}^{n}\cross \mathbb{C}_{\tau}$.
With these coordinates, we often identify $TyX$ and $T_{\mathrm{Y}}L$ with $X$ and $L$ respectively.
The projection $\tau_{\mathrm{Y}}$: $T_{\mathrm{Y}}Larrow \mathrm{Y}$ and $s_{L}$:
$T_{\mathrm{Y}}Larrow\overline{L}_{\mathrm{Y}}$ induce natural mappings:
$T_{N}^{*} \mathrm{Y}T_{N}M\cross T_{N}^{*}\mathrm{Y}\overline{\tau_{Y\pi}}N\vec{\ell_{\mathcal{T}_{\acute{Y}}}}\overline{\overline{{}^{t}s_{\acute{L}}}}-T_{T_{N}hI}^{*}T_{\mathrm{Y}}LT_{N}MT\frac{*}{\mathrm{A}I}\tilde{L}_{\mathrm{Y}}\frac{\cross}{M}NN\vec{s_{L\pi}}NT\frac{*}{M}\tilde{L}_{\mathrm{Y}}$,
and by these mappings, we identify $T_{T_{N}h\mathrm{f}}^{*}T_{\mathrm{Y}}L$ with $T_{N}M\cross T_{N}^{*}\mathrm{Y}N$ and $T_{N}MT \frac{*}{M}\frac{\cross}{M}N\tilde{L}_{\mathrm{Y}}N$
$T_{\mathrm{Y}}L\backslash T_{\mathrm{Y}}\mathrm{Y}$ has two components with respect to its fiber. We denote one of them by $T_{\mathrm{Y}}L^{+}$ and represent (at least locally) by fixing
an
admissible coordinate system$T_{\mathrm{Y}}L^{+}=\{(z, t)\in T_{\mathrm{Y}}L;t>0\}$ .
Moreover set $T_{N}M^{+}:=T_{\mathrm{Y}}L^{+}\cap T_{N}M$
.
Set an open embedding $f:T_{\mathrm{Y}}L^{+}\mapsto T_{\mathrm{Y}}L$ and$f_{N}:=f|_{T_{N}M^{+}}:$ $T_{N}M^{+}\mapsto T_{N}M$. We regard $T_{N}M^{+}\cross T_{N}^{*}\mathrm{Y}N$ as an open set of$T_{T_{N}M}^{*}T_{\mathrm{Y}}L$.
Moreover $f$ induces mappings:
$T_{T_{N}M^{+}}^{*}T_{\mathrm{Y}}L^{+}--T_{N}M^{+}\cross T_{T_{N}M}^{*}T_{\mathrm{Y}}Larrow T_{T_{N}M}^{*}T_{\mathrm{Y}}LT_{N}Mf_{n}$
$|^{\iota}$
$\mathrm{O}$
$|l$
$T_{N}M^{+}\cross T_{N}^{*}\mathrm{Y}N$ $N\cross T_{N}^{*}\mathrm{Y}$
.
Hence
we
identify $T_{T_{N}M}^{*}+T_{\mathrm{Y}}L^{+}$ with $T_{N}M^{+}\cross T_{N}^{*}\mathrm{Y}N$’and
$f_{\pi}$ with $f_{N}\cross \mathrm{i}\mathrm{d}$.
Let $\pi_{N,M}$: $T_{\frac{*}{M}N}\tilde{L}_{\mathrm{Y}}arrow\overline{M}_{N}$ and $\pi_{N|M}$: $T_{T_{N}M}^{*}T_{\mathrm{Y}}Larrow T_{N}M$, be the natural projections.
We denote
as
usual by $\nu$ and $\mu$ the Sato specialization and microlocalization functorsrespectively.
2General Boundary
Values
By using an admissible coordinate system we define acontinuous section $\sigma:\mathrm{Y}arrow\dot{T}_{\mathrm{Y}}X$
by $z-\rangle$ $(z, 1)$
.
Similarly we define ${}^{t}\sigma:\mathrm{Y}arrow\dot{T}_{\mathrm{Y}}^{*}X$ by $z\vdash\Rightarrow(z, 1)$.
In general, let $Z$ beacomplex manifold, $\tau:Earrow Z$ acomplex vector bundle. Then, denote by $\mathrm{D}_{\mathbb{C}^{\mathrm{x}}}^{b}(E)$ the
subcategory of$\mathrm{D}^{b}(E)$ consisting of $\mathbb{C}^{\mathrm{x}}$-conic objects.
2.1 Theorem. For any object $\mathcal{F}$
of
$\mathrm{D}^{b}(X)$ such that $\nu_{\mathrm{Y}}(\mathcal{F})\in \mathrm{O}\mathrm{b}(\mathrm{D}_{\mathbb{C}^{\mathrm{x}}}^{b}(T_{\mathrm{Y}}X))$ , thereexists the following natural isomorphism:
$f_{\pi}^{-1}\mu_{T_{N}M(\nu_{\mathrm{Y}}(i_{i’}\mathcal{F}))3f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mu_{N}(\sigma^{-1}\nu_{\mathrm{Y}}(\mathcal{F}))\otimes\omega_{L/X}}$
.
2.2 Definition. Foranyobject $\mathcal{F}$of$\mathrm{D}^{b}(X)$ such that $\nu_{\mathrm{Y}}(\mathcal{F})\in \mathrm{O}\mathrm{b}(\mathrm{D}_{\mathbb{C}^{\mathrm{X}}}^{b}(T_{\mathrm{Y}}X))$,we define
by virtue of Kashiwara-Schapira [K-S] and Theorem 2.1:
$\beta:f_{\pi}^{-1}s_{L\pi}^{-1}\mu_{\overline{M}_{N}}(Rj_{L*}\tilde{p}_{L}^{-1}i_{i’}\mathcal{F})arrow f_{\pi}^{-1}\mu_{T_{N}M}(\nu_{\mathrm{Y}}(i_{i’}\mathcal{F}))$
$\approx$ $f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mu_{N}(\sigma^{-1}\nu_{\mathrm{Y}}(\mathcal{F}))\otimes\omega_{L/X}$
.
2.3 Definition (Laurent-Monteiro Fernandes $[\mathrm{L}$-MF2]). We say
an
object $\mathcal{F}$ of$\mathrm{D}^{b}(X)$ is near-hyperbolic at $x_{0}\in N$ (in $dt$-codirection)ifthere exist positive constants $C$
and $\epsilon_{1}$ such that
$\mathrm{S}\mathrm{S}(\mathcal{F})\cap\{(z, \tau;z^{*}, \tau^{*})\in T^{*}X;|z-x_{0}|, |\tau|<\epsilon_{1},0<{\rm Re}\tau\}$
$\subset\{(z, \tau;z^{*}, \tau^{*})\in T^{*}X;|{\rm Re}\tau^{*}|<C(|{\rm Im} z^{*}|(|{\rm Im} z|+|{\rm Im}\tau|)+|{\rm Re} z^{*}|)\}$
holds by an admissible coordinatesystem. Here $\mathrm{S}\mathrm{S}(\mathcal{F})$ denotes the microsupport of J.
2.4 Theorem. Let $\mathcal{F}$ be a object
of
$\mathrm{D}^{b}(X)$.
Assume that $\nu_{\mathrm{Y}}(\mathcal{F})\in \mathrm{O}\mathrm{b}(\mathrm{D}_{\mathbb{C}^{\mathrm{x}}}^{b}(T_{\mathrm{Y}}X))$ and$\mathcal{F}$ is near-hyperbolic at
$x_{0}\in N$. Then,
for
any$p^{*}\in T_{T_{N}M^{+}}^{*}T_{\mathrm{Y}}L^{+}$$\beta$: $s_{L\pi}^{-1}\mu_{\overline{M}_{N}}(Rj_{L*}\tilde{p}_{L}^{-1}i_{i’}\mathcal{F})_{P^{*}}$ $arrow\mu_{N}(\sigma-1\nu_{\mathrm{Y}}(\mathcal{F}))_{\tau_{Y\pi}(p*)}\otimes\omega_{L/X}$
is an isomorphism.
3Regular-Specializable Systems
In this section, we shall recall the basic results concerning the regular-specializable $\mathcal{D}-$
Module and its nearby-cycle.
As usual, we denote by $\mathcal{D}_{X}$ the sheafon $X$ ofholomorphic differential operators, and
by $\{\mathcal{D}_{X}^{(m)}\}_{m\in \mathrm{N}_{0}}$ the usual order filtration on $\mathcal{D}_{X}$ .
3.1 Definition. Denote by $\mathrm{J}_{\mathrm{Y}}$ the defining Ideal of
$\mathrm{Y}$ in
$O_{X}$ with aconvention that $0_{\mathrm{Y}}^{j}=(9_{X}$ for $j\leq 0$. The $V$
filtration
$\{V_{\mathrm{Y}}^{k}(\mathcal{D}_{X})\}_{k\in \mathbb{Z}}$ (along Y) is afiltration on $\mathcal{D}_{X}|_{\mathrm{Y}}$defined by
$V_{\mathrm{Y}}^{k}( \mathcal{D}_{X}):=\bigcap_{j\in \mathbb{Z}}\{P\in \mathcal{D}_{X}|_{\mathrm{Y}}; P0_{\mathrm{Y}}^{j}\subset \mathrm{J}_{\mathrm{Y}}^{j-k}\}$.
Let us denote by 19 the Euler operator. Note that $\theta\in V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})\backslash V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})$ and that $\theta$
can be represented by $\tau\partial_{r}$ by admissible coordinates.
3.2 Definition. Acoherent $\mathcal{D}_{X}|_{\mathrm{Y}}$-Module $\mathrm{M}$ is said to be regular-specializable (along
Y) if there exist locally acoherent $\mathrm{t}9_{X}$-sub-Module $\mathrm{M}_{0}$ of $\mathrm{M}$ and anon-zero polynomial
$b(\alpha)\in \mathbb{C}[\alpha]$ such that the following conditions are satisfied:
(1) $\mathrm{M}_{0}$ generates $\mathrm{M}$ over $\mathcal{D}_{X}$ ; that is, $\mathrm{M}$ $=\mathcal{D}_{X}\mathrm{M}_{0}$ ;
(2) $b(\theta)\mathrm{M}_{0}\subset(\mathcal{D}_{X}^{(m)}\cap V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X}))\mathrm{M}_{0}$, where $m$ is the degree of$b(\alpha)$.
In what follows, we shall omit the phrase “along $\mathrm{Y}$”since $\mathrm{Y}$ is fixed.
3.3 Remark. (1) Let $\mathrm{M}$ be acoherent $\mathcal{D}_{X}|_{\mathrm{Y}}$-Module for which $\mathrm{Y}$ is non-characteristic.
Then, it is easy to see that $\mathrm{M}$ is regular-specializable.
(2) Kashiwara-Kawai [K-K] proved that every regular-holonomic $\mathcal{D}_{X}|_{\mathrm{Y}}$-Module is
regular-specializable.
3.4 Proposition.
If
M is a regular-specializable $\mathcal{D}_{X}|_{\mathrm{Y}}$-Module, $R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mu_{\mathrm{Y}}(0_{X}))$and $R\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \nu_{\mathrm{Y}}(O_{X}))$ are objects
of
$\mathrm{D}_{\mathbb{C}^{\mathrm{X}}}^{b}(T_{\mathrm{Y}}^{*}X)$ and $\mathrm{D}_{\mathbb{C}^{\cross}}^{b}(T_{\mathrm{Y}}X)$ respectively.Let $\iota:\mathrm{Y}arrow X$ be the natural inclusion. Then the inducedsystem, or the inverse image
in the sense of $\mathcal{D}$-Modules is defined by $D\iota^{*}\mathrm{M}$
$:=\mathrm{t}9_{\mathrm{Y}}\otimes^{L}\iota^{-1}\iota^{-1}0_{X}$M.
For any regular-specializable $\mathcal{D}_{X}$-Module $\mathrm{M}$, the nearby-cycle $\Psi_{\mathrm{Y}}(\mathrm{M})$ of $\mathrm{M}$ and the
vanishing-cycle $\Phi_{\mathrm{Y}}(\mathrm{M})$ of $\mathrm{M}$ in the theory of $\mathcal{D}$-Modules can be defined. For the
defini-tions of$\Psi_{\mathrm{Y}}(\mathrm{M})$ and $\Phi_{\mathrm{Y}}(\mathrm{M})$, we refer to Laurent [L], Mebkhout [Me]. We shall recall the
following two results
3.5 Proposition (Laurent [L], Mebkhout [Me]). Let M be a regular-specializable
$D_{X}|_{\mathrm{Y}}$-Module. Then, ’$\mathrm{Y}(’ \mathrm{C})_{\rangle}I_{\mathrm{Y}}(\ovalbox{\tt\small REJECT} M)$ and each cohomology
of
Dc’M are coherent $\ovalbox{\tt\small REJECT} \mathrm{j})_{Y}$ Modules. Moreover, there exists the following distinguished triangle:$\Phi_{\mathrm{Y}}(\mathrm{M})$
Var
$\Psi_{\mathrm{Y}}(\mathrm{M})arrow D\iota^{*}\mathrm{M}$ $arrow^{+1}$ . $/fere$, Var $:=\varphi(\theta)\tau$ with $\varphi(\zeta)$ $:=(e^{2\pi\sqrt{-1}\zeta}-1)/\zeta$.
3.6 Theorem (Laurent [L]). Let $\mathrm{e}_{\mathrm{Y}|X}^{\mathrm{R}}$ be the
sheaf of
real holomorphicmicrofunctions
on
$T_{\mathrm{Y}}^{*}X$as
usual. Let $\mathrm{M}$ be a regular-specializable $\mathcal{D}_{X}|_{\mathrm{Y}}$-Module. Then, there exists thefollowing isomorphism
of
distinguished triangles:$R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{t}9_{X})|_{\mathrm{Y}}arrow R\}(om_{\mathrm{D}_{X}}(\mathrm{M},\sigma^{-1}\nu_{\mathrm{Y}}(\mathfrak{l}9_{X}))arrow Mom_{\mathrm{D}_{X}}(\mathrm{M},{}^{t}\sigma^{-1}\mathrm{G}_{\mathrm{Y}|X}^{\mathbb{R}})arrow+1$
$\downarrow l$ $\downarrow[$ $\downarrow l$
$Rg[om_{\mathrm{D}_{Y}}(D\iota^{*}\mathrm{M}, 19_{\mathrm{Y}})arrow R\sigma(om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), 19_{\mathrm{Y}})arrow Mom_{\mathrm{D}_{Y}},(\Phi_{\mathrm{Y}}(\mathrm{M}), \mathrm{t}9_{\mathrm{Y}})arrow+1$
3.7 Remark. (1) The isomorphism (the Cauchy-Kovalevskaja type theorem)
$R\mathcal{H}om_{\mathrm{D}_{Y}}(D\iota^{*}\mathrm{M}, \mathrm{t}9_{\mathrm{Y}})\simeq R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{t}9_{X})|_{\mathrm{Y}}$
holds for Fuchsian systems in the
sense
of Laurent-Monteiro Fernandes [$\mathrm{L}$-MF 1].(2) Recently Mandai [Man] extended the definition of boundary values to ageneral
Fuchsian differential equation in the complex domain.
4Boundary Values
for Regular-Specializable
System
We denote by Ox, $\mathfrak{B}_{M}$and $\mathrm{e}_{M}$ thesheaf ofholomorphic
functions
on
$X$, ofhyperfunctionson
$M$ and ofmicrofunctions
on
$T_{M}^{*}X$ respectively.4.1 Definition (Oaku [Oa 2], Oaku-Yamazaki [O-Y]). We set:
$\mathrm{G}_{N|M}:=s_{L\pi}^{-1}\mu_{\overline{M}_{N}}(Rj_{L*}\tilde{p}_{L}^{-1}ii’\mathrm{t}9_{X})\otimes or_{M/X}[n+1]$
.
We
can
regard $\mathrm{e}_{N|M}$as
amicrolocalization of $\nu_{N}(\mathfrak{B}_{M})$:4.2 Proposition. (1) $\mathrm{e}_{N|M}$ is concentrated in degree zero; that is, $\mathrm{e}_{N|M}$ is regarded as $a$
sheaf
on $T_{T_{N}M}^{*}T_{\mathrm{Y}}L$.
Further $\mathrm{G}_{N|M}|_{T_{N}M}=\nu_{N}(\mathfrak{B}_{M})$ holds(2) There exists the following exact sequence on $T_{N}M$:
$0arrow\nu_{\mathrm{Y}}(\mathfrak{B}\mathrm{t}9_{L})|_{T_{N}M}arrow\nu_{N}(\mathfrak{B}_{M})arrow\dot{\pi}_{N|M*}\mathrm{C}_{N|M}arrow 0$
.
Here$\mathfrak{B}\mathrm{t}9_{L}:=\mathcal{H}_{L}^{1}(\mathrm{t}9_{X})\otimes or_{L/X}$ is the
sheaf
of
hyperfunctions with holomorphic parameterson L. Note that $\nu_{\mathrm{Y}}(\mathfrak{B}\mathrm{t}9_{L})$ is concentrated in degree zero;
4.3 Definition. Let M be aregular-specializable $\mathrm{D}_{X}|_{\mathrm{Y}}$-Module. By Proposition 3.4, $R\ovalbox{\tt\small REJECT} J^{-}Com$
.
(M,$\mathrm{C}9_{X})$ satisfies the assumption of Theorem 2.1. Thus, by Definition 2.2 and$x$
Theorem 3.6, we define:
$\beta:f_{\pi}^{-1}R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{G}_{N|M})arrow f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}R\mathrm{H}om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), \mathrm{G}_{N})$.
4.4 Theorem. (1) The morphism $\beta$ gives a monomorphism
$\beta^{0}$: $f_{\pi}^{-1}\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{C}_{N|M})\mapsto f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mathcal{H}om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), \mathrm{C}_{N})$ .
(2) The restriction
of
$\beta^{0}$ to the zerO-section $T_{N}M^{+}$ coincides with the boundary valuemorphism in the sense
of
MonteiroFernandes
$[\mathrm{M}\mathrm{F}1]$.4.5 Definition. Let $\mathrm{M}$ be acoherent $\mathcal{D}_{X}|_{\mathrm{Y}}$-Module. Then we say $\mathrm{M}$ is near-hyperbolic
at $x_{0}\in N$ (in $dt$-codirection)if $R9\{om_{\mathrm{D}_{X}},(\mathrm{M}, 0_{X})$ is near-hyperbolic in the
sense
ofDefinition 2.3. Here, we remark that $\mathrm{S}\mathrm{S}(R\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \mathrm{t}9_{X}))=\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(\mathrm{M})$.
The following theorem is adirect consequence ofTheorem 2.4:
4.6 Theorem. Let $\mathrm{M}$ be a regular-specializable $\mathcal{D}_{X}|_{\mathrm{Y}}$-Module. Assume that $\mathrm{M}$ is
near-hyperbolic at $x_{0}\in N$
.
Then,for
any $p^{*}\in T_{T_{N}M^{+}}^{*}T_{\mathrm{Y}}L^{+}$$\beta:R\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \mathrm{G}_{N|M})_{p^{*}}arrow R\mathcal{H}om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), \mathrm{C}_{N})_{\tau_{Y\pi}(p*)}$
is an isomorphism.
4.7 Remark. Let $\mathrm{C}_{N|M}^{F}$ be the sheaf of $F$-mild microfunctions on $T_{T_{N}M}^{*}T_{\mathrm{Y}}L$, and set
$\overline{\mathrm{e}}_{N|M}^{A}:=\mathcal{H}^{n}(\mu_{N}(\mathit{0}_{X}|_{\mathrm{Y}}))\otimes or_{N/\mathrm{Y}}$(see Oaku [Oa 1], [Oa 2], and Oaku-Yamazaki [O-Y]).
Let $\mathrm{M}$ be aregular-specializable $\mathcal{D}_{X}|_{\mathrm{Y}}$-Module. Set $\mathrm{M}_{\mathrm{Y}}:=\mathcal{H}^{0}(D\iota^{*}\mathrm{M})$ $=\mathrm{t}9_{\mathrm{Y}}$
$\iota^{-1}0_{X}\otimes\iota^{-1}$M.
By the argument in Oaku-Yamazaki [O-Y] we have the following commutative diagram:
$f_{\pi}^{-1} \mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{G}_{N|M}^{F})\approx f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mathcal{H}om_{\mathrm{q})_{X}}(\mathrm{M},\tilde{\mathrm{C}}_{N|M}^{A})\int 0\downarrow\backslash -\vec{\mathrm{o}}\iota’f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mathrm{H}om_{\mathrm{D}_{Y}}(\mathrm{M}_{\mathrm{Y}}, \mathrm{C}_{N})$
$f_{\pi}^{-1}\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \mathrm{G}_{N|M})>arrow f_{\pi}^{-1}\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M},\overline{\mathrm{G}}_{N|M})arrow-f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}f\mathrm{f}om_{\mathrm{D}_{Y}}(\Psi_{\mathrm{Y}}(\mathrm{M}), \mathrm{C}_{N})$,
that is, the boundary value morphism
$\gamma^{F}$: $f_{\pi}^{-1}\mathrm{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{G}_{N|M}^{F})\mapsto f_{\pi}^{-1}\tau_{\mathrm{Y}\pi}^{-1}\mathrm{f}\{om_{\mathrm{D}_{Y}}(\mathrm{M}_{\mathrm{Y}}, \mathrm{C}_{N})$
and $\beta^{0}$ are compatible. In particular, if $\mathrm{Y}$ is non-characteristic for $\mathrm{M}$, then it is known
that $\Psi_{\mathrm{Y}}(\mathrm{M})\approx$ $D\iota^{*}\mathrm{M}$ $\simeq \mathrm{M}_{\mathrm{Y}}$ and by Oaku [Oa2] (cf. Oaku-Yamazaki [O-Y]) we have $\overline{\gamma}_{N|M}$: $R\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M},\overline{\mathrm{C}}_{N|M})\approx$ $\tau_{\mathrm{Y}\pi}^{-1}R\mathcal{H}om_{\mathrm{D}_{Y}}(\mathrm{M}_{\mathrm{Y}}, \mathrm{G}_{N})$.
In thiscase we seethat$\beta^{0}$is equivalent to the non-characteristic boundary
valuemorphism
(seeKataoka [Kat] and Oaku [Oa 2]). Inparticular, the restriction of$\beta^{0}$ tothe zer0-section
$T_{N}M^{+}$ is equivalent to Komatsu-Kawai [KO-K] and Schapira [Sc 1]. Further, if$\mathrm{Y}$ is
non-characteristic for $\mathrm{M}$ and $\pm dt\in T_{N}^{*}M$ is hyperbolic for $\mathrm{M}$, then the nearly-hyperbolic
condition is satisfied and $\beta$ is
an
isomorphism.4.8 Example. Assume that $X=\mathbb{C}^{n+1}$ and
so
on byan
admissible coordinate system.(1) Let $b(\alpha)$ be
anon-zero
polynomial with degree $m$, and $Q\in \mathcal{D}_{X}^{(m)}\cap V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})$.Set $\mathrm{M}$ $:=\mathcal{D}_{X}/\mathcal{D}_{X}(b(\theta)+Q)$. Then $\mathrm{M}$ is regular-specializable. Assume
that $\mathrm{b}(\mathrm{a})=$
$\prod_{j=1}^{\mu}(\alpha-\alpha_{j})^{\nu_{\mathrm{j}}}$ ($\alpha_{i}-\alpha_{j}\not\in \mathbb{Z}$ for $1\leq i\neq j\leq\mu$, note that $\sum_{j=1}^{\mu}\nu_{j}=m$). Then adirect
calculation shows that $\Psi_{\mathrm{Y}}(\mathrm{M})\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus m}$, and$\beta^{0}$ is equivalent to
7in
Oaku [Oa2]: Let$p^{*}=$$(x_{0}, t_{0};\sqrt{-1}\langle\xi_{0}, dx\rangle)$ be apoint of
$T_{T_{N}M}^{*}+T_{\mathrm{Y}}L^{+}$, and $f(x,t)$ agerm of $\mathcal{H}om_{\mathrm{D}_{X}},(\mathrm{M}, \mathrm{C}_{N|M})$
at $p^{*}$
.
Then,we can
see that $f(x, t)$ has adefining function$F(z, \tau)=\sum_{j=1}^{\mu}\sum_{k=1}^{\nu_{j}}F_{jk}(z, \tau)\tau^{\alpha_{\mathrm{j}}}(\log\tau)^{k-1}$
.
Here each$F_{jk}(z, \tau)$ is holomorphic
on
aneighborhood of$\{(z, \mathrm{O})\in X;|x_{0}-z|<\epsilon$, ${\rm Im} z\in$$\Gamma\}$ with apositive constant$\epsilon$ and
an
openconvex cone
$\Gamma$ such that$\xi_{0}\in \mathrm{I}\mathrm{n}\mathrm{t}(\mathrm{I}^{\mathrm{o}})$ (theinte-rior of the dual
cone
$\Gamma^{\mathrm{o}}$ of$\Gamma$). Then, $\beta^{0}(f)$ isequivalent to$\{\mathrm{s}\mathrm{p}_{N}(F_{jk}(x+\sqrt{-1}\Gamma 0,0));1\leq$
$k\leq\nu_{j}$, 1 $\leq j\leq\mu$
}.
Moreover, if the principal symbol of $b(\theta)+Q$ is written as$\tau^{m}P(z, \tau;z^{*}, \tau^{*})$ for ahyperbolic polynomial $P$ at $dt$-codirection, the nearly-hyperbolic
condition is satisfied. Note that this operator is aspecial
case
of Fuchsian hyperbolicoperators due to Tahara [T].
(2) Take
an
operator $A(z;\partial_{z})\in \mathcal{D}_{\mathrm{Y}}^{(1)}$ at the origin and set $A^{0}:=\mathrm{i}\mathrm{d}$ and $A^{(j)}:=$$\frac{1}{j!}A\circ A^{(j-1)}\in \mathcal{D}_{\mathrm{Y}}^{(j)}$ for $j\geq 1$
.
Let $p^{*}=(0,1;\sqrt{-1}\langle\xi, dx\rangle)$ be apoint of$T_{T_{N}M^{+}}^{*}T_{\mathrm{Y}}L^{+}$ and
set $p_{0}:=(0;\sqrt{-1}\langle\xi, dx\rangle)\in T_{N}^{*}\mathrm{Y}$
.
Set $P:=(\theta-\alpha_{1})(\theta-\alpha_{2})-\tau A(z;\partial_{z})\theta\in \mathcal{D}_{X}|_{\mathrm{Y}}$ , where$(\alpha_{1}, \alpha_{2})\in \mathbb{C}^{\oplus 2}$
.
Consider $\mathrm{M}$ $:=\mathcal{D}_{X}/\mathcal{D}_{X}P=\mathcal{D}_{X}u$, where $u:=1\mathrm{m}\mathrm{o}\mathrm{d} P$.
Let$f(x, t)$ be
agerm of$\mathcal{H}om_{\mathrm{D}_{X}}(\mathrm{M}, \mathrm{C}_{N|M})$ at $p^{*}$
.
Then:(i) If $(\alpha_{1}, \alpha_{2})=(-1,0)$, then
$\Phi_{\mathrm{Y}}(\mathrm{M})=\frac{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})(\theta+1)u}{V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})(\theta+1)u}=\mathcal{D}_{\mathrm{Y}}[u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}(\theta+1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$ ,
$\Psi_{\mathrm{Y}}(\mathrm{M})=\frac{V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})(\theta+1)u}{V_{\mathrm{Y}}^{-2}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})(\theta+1)u}=\mathcal{D}_{\mathrm{Y}}[\tau u]+\mathcal{D}_{\mathrm{Y}}[(\theta+1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$,
and Var: $([u], [\partial_{\tau}(\theta-1)u])-+([\tau u], 0)$
.
Hence $\mathrm{M}_{\mathrm{Y}}\simeq \mathcal{D}_{\mathrm{Y}}[(\theta+1)u]\simeq \mathcal{D}_{\mathrm{Y}}$.
In this case$f(x,t)$ has the following defining function
$F(z, \tau)=U_{0}(z)+\frac{U_{-1}(z)}{\tau}-\sum_{j=1}^{\infty}\frac{A^{(j)}U_{-1}(z)}{j-1}\tau^{j-1}-AU_{-1}(z)\log\tau$ ,
and
!
$(f^{\ovalbox{\tt\small REJECT}}(\ovalbox{\tt\small REJECT} \mathrm{m}\ovalbox{\tt\small REJECT}, t))$ is given by $\{\mathrm{s}\mathrm{p}_{N}(U_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}})(\mathrm{r})\}_{*\ovalbox{\tt\small REJECT}}\mathrm{i},0$ at$p_{\mathit{0}}$ . If
$f(\mathrm{m}\ovalbox{\tt\small REJECT}, [])$ is $F$-mild at Po\rangle then
$U.(z)\ovalbox{\tt\small REJECT}$ 0 and $\mathrm{t}^{F}(f(x, t))\ovalbox{\tt\small REJECT}$ $\{f(\mathrm{r}, +\mathrm{O})\}\ovalbox{\tt\small REJECT}$ $\{\mathrm{s}\mathrm{p}_{N}(U_{0})(\mathrm{z})\}_{\ovalbox{\tt\small REJECT}}$
(ii) If C’r:’2) $\ovalbox{\tt\small REJECT}$ $(0_{\ovalbox{\tt\small REJECT}}1)_{\ovalbox{\tt\small REJECT}}$ then
$\Phi_{\mathrm{Y}}(\mathrm{M})$ $= \frac{V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})\theta u}{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})\theta u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}\theta u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$ ,
$\Psi_{\mathrm{Y}}(\mathrm{M})$ $= \frac{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})\theta u}{V_{\mathrm{Y}}^{-1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})\theta u}=\mathcal{D}_{\mathrm{Y}}[u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}\theta u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$,
and Var$[\partial_{\tau}u]=\mathrm{V}\mathrm{a}\mathrm{r}$$[\partial_{\tau}^{2}\theta u]=0$
.
Hence $\mathrm{M}_{\mathrm{Y}}\simeq \mathcal{D}_{\mathrm{Y}}[u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}\theta u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$ In thiscase
$f(x, t)$ has the following defining function:
$F(z, \tau)=U_{0}(z)+\sum_{j=0}^{\infty}\frac{A^{(j)}U_{1}(z)}{j+1}\tau^{j+1}$,
and $f(x, t)$ is always $F$-mild. Hence $\beta^{0}(f(x, t))$ at $p_{0}$ coincides with $\gamma^{F}(f(x, t))=$
$\{\partial_{t}^{i}f(x, +0)\}_{i=0,1}=\{\mathrm{s}\mathrm{p}_{N}(U_{i})(x)\}_{i=0,1}$ (if $\tau\neq 0$, $\mathrm{M}$ is isomorphic to $\mathcal{D}_{X}/\mathcal{D}_{X}(\partial_{\tau}^{2}$
-$A(z;\partial_{z})\partial_{\tau})$ for which $\mathrm{Y}$ is non-characteristic).
(iii) If $(\alpha_{1}, \alpha_{2})=(1,1)$, then
$\Phi_{\mathrm{Y}}(\mathrm{M})$ $= \frac{V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})u}{V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}(\theta-1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$,
$\Psi_{\mathrm{Y}}(\mathrm{M})$ $= \frac{V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})u}{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}(\theta-1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$
and Var: $([\partial_{\tau}^{2}u], [\partial_{\tau}^{2}(\theta-1)u])-t(2\pi\sqrt{-1}[\partial_{\tau}(\theta-1)u], 0)$. Hence $\mathrm{M}_{\mathrm{Y}}\simeq \mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]\simeq \mathcal{D}_{\mathrm{Y}}$.
In this case $f(x, t)$ has the following defining function:
$F(z, \tau)=\sum_{j=0}^{\infty}A^{(j)}U_{0}(z)\tau^{j+1}-\sum_{j=1}^{\infty}\sum_{k=1}^{j}\frac{A^{(j)}U_{1}(z)}{k}\tau^{j+1}+\sum_{j=0}^{\infty}A^{(j)}U_{1}(z)\tau^{j+1}\log\tau$ ,
and $\beta^{0}(f(x, t))$ is given by $\{\mathrm{s}\mathrm{p}_{N}(U_{i})(x)\}_{i=0,1}$ at $p_{0}$ . If $f(x, t)$ is $F$-mild at $p_{0}$, then $U_{0}(z)=0$ and $\gamma^{F}(f(x, t))=\{\partial_{t}f(x, +0)\}=\{\mathrm{s}\mathrm{p}_{N}(U_{1})(x)\}$.
(iv) If $(\alpha_{1}, \alpha_{2})=(1,2)$, then:
$\Phi_{\mathrm{Y}}(\mathrm{M})$ $= \frac{V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{3}(\mathcal{D}_{X})(\theta-1)u}{V_{\mathrm{Y}}^{1}(2)_{X})u+V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})(\theta-1)u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{3}(\theta-1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$,
$\Psi_{\mathrm{Y}}(\mathrm{M})$ $= \frac{V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{2}(\mathcal{D}_{X})(\theta-1)u}{V_{\mathrm{Y}}^{0}(\mathcal{D}_{X})u+V_{\mathrm{Y}}^{1}(\mathcal{D}_{X})(\theta-1)u}=\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}(\theta-1)u]\simeq \mathcal{D}_{\mathrm{Y}}^{\oplus 2}$ ,
and Var: $([\partial_{\tau}^{2}u], [\partial_{\tau}^{3}(\theta-1)u])\vdash\Rightarrow(0,2A[\partial_{\tau}u])$. Hence
$\mathrm{M}_{\mathrm{Y}}\simeq\frac{\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}u]+\mathcal{D}_{\mathrm{Y}}[\partial_{\tau}^{2}(\theta-1)u]}{\mathcal{D}_{\mathrm{Y}}A[\partial_{\tau}u]}$ .
In this case $f(x, t)$ has the following defining function:
$F(z, \tau)=\sum_{j=0}^{\infty}A^{(j)}U_{2}(z)\tau^{j+2}+U_{1}(z)\tau-\sum_{j=2}^{\infty}\sum_{k=1}^{j-1}\frac{jA^{(j)}U_{1}(z)}{k}\tau^{j+1}$
$+( \sum_{j=0}^{\infty}(j+1)A^{(j+1)}U_{1}(z)\tau^{j})\tau^{2}\log\tau$,
and $\beta^{0}(f(x, t))$ is given by $\{\mathrm{s}\mathrm{p}_{N}(U_{\dot{1}})(x)\}_{*=1,2}$. at $p_{0}$
.
$f(x, t)$ is $F$-mild under thecon-dition that $AU_{1}(z)=0$, and in this
case
$\gamma^{F}(f(x, t))$ at $p_{0}$ is given by $\gamma^{F}(f_{3}(x, t))=$$\{\partial_{t}^{:}f(x, +0)\}_{:=1,2}=\{\mathrm{s}\mathrm{p}_{N}(U_{1})(x), 2\mathrm{s}\mathrm{p}_{N}(U_{2})(x)\}$ with $A\partial_{t}f(x, +0)=A\mathrm{s}\mathrm{p}_{N}(U_{1})(x)=0$.
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